by Derelle McFarland

Thanks to Euler and Feuerbach we have a way to define and compose a circle other than by midpoint and radius, by what we call the nine point circle.
The nine point circle is also called the Euler Circle and the Feuerbach Circle, because both mathematicians contributed to the circle.
I will show you how to construct the nine point circle. First given triangle ABC, construct the altitudes of each side. Then construct a triangle that connects the points of intersection between the sides of the triangle and the altitudes.

Then construct the midpoints of each side of the triangle. Then construct a triangle from these three points.

Now construct the orthocenter and construct segments from the orthocenter to each vertex. Then construct the midpoint of these segments. Then connect the midpoints to form a third triangle inside the original triangle.

Now we can connect all of the points except for the orthocenter to construct the nine point circle.

You can construct the nine point circle from any triangle.
The following is the construction of the nine point circle from an equilateral triangle.

It appears that there are only six point, but in actuality the blue/red triangle is really two triangles on top of each other. It is obvious that the blue triangle is the construction of a triangle from the midpoints of the original triangle, but the red triangle is at the same place because the altitude of an equilateral triangle goes through the midpoint of the sides.

An obtuse triangle is a little trickier. The construction of the red triangle which is the midpoints of the original triangle is simple to construct. The orthocenter of an obtuse triangle is outside of the triangle. So we must continue the sides of the triangles so that we can find the intersection between the altitudes and the sides of the triangle. The constructio will follow.

As you see this is a very interesting construction of a circle which requires students to know other parts of triangles like altitudes and medians. It could be very useful in a geometry class.

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